Inverse Variation Calculator

Inverse variation describes a relationship in which one quantity increases while another decreases so that their product remains constant. The standard equation is y = k divided by x, where x and y are the changing variables and k is the constant of variation. Multiplying both sides by x gives k = x times y. Once one matching pair is known, that product identifies the entire inverse relationship and lets you calculate any other matching pair. This pattern differs from direct variation. In direct variation, y divided by x stays constant and doubling x doubles y. In inverse variation, x times y stays constant and doubling x cuts y in half. Tripling x reduces y to one third of its previous value. The graph is a hyperbola rather than a straight line, and it approaches both coordinate axes without touching them when the constant is nonzero. Recognizing which quantity remains constant is the key to choosing the correct model. The calculator first multiplies the initial x and y values to obtain k. It then divides k by the new x value to find the corresponding new y value. For example, if x is 2 when y is 12, the constant is 24. When x changes to 8, y becomes 24 divided by 8, or 3. The products 2 times 12 and 8 times 3 are equal, confirming that both points belong to the same inverse variation. Inverse proportionality appears in many practical situations. At a fixed distance, travel time varies inversely with average speed. For a fixed amount of work under ideal assumptions, completion time can vary inversely with the number of equally productive workers. Gas pressure and volume follow an inverse relationship at constant temperature in Boyle's law. The wavelength of a wave is inversely related to frequency when propagation speed remains constant. Gear speed and tooth count also provide familiar mechanical examples. Zero cannot be used for either x value because division by zero is undefined. Negative values are mathematically valid and produce a hyperbola in the opposite quadrants depending on the sign of k, though a real application may restrict quantities such as time or distance to positive values. Decimal inputs are accepted, and results are rounded only for display. Always check that inverse variation is a reasonable model for the underlying situation before interpreting the calculated value.